Silverman, Joseph H.; Tate, John Rational points on elliptic curves. (English) Zbl 0752.14034 Undergraduate Texts in Mathematics. New York: Springer-Verlag. x, 281 p. (1992). The book gives a good introduction for students which are interested in Diophantine equations and arithmetic geometry. It is based on lectures of J. Tate from 1961. It contains a lot of exercises. Often further developments and applications are explained, for instance Lenstra’s algorithm for factorisation of integers using elliptic curves.The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. — At the end one finds complex multiplication and Galois representations associated to torsion points.The algebraic geometry needed for the purpose of the book (for instance Bézout’s theorem) is presented in an appendix. Reviewer: Gerhard Pfister (Berlin) Cited in 9 ReviewsCited in 111 Documents MSC: 11G05 Elliptic curves over global fields 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 11G15 Complex multiplication and moduli of abelian varieties 11D25 Cubic and quartic Diophantine equations 14G05 Rational points 14G15 Finite ground fields in algebraic geometry 14H52 Elliptic curves Keywords:Diophantine equations; elliptic curves; Nagell-Lutz theorem; Mordell-Weil theorem; rational points; Thue-Siegel theorem; integer points; finite fields; complex multiplication; torsion points PDF BibTeX XML Cite \textit{J. H. Silverman} and \textit{J. Tate}, Rational points on elliptic curves. New York: Springer-Verlag (1992; Zbl 0752.14034) OpenURL Digital Library of Mathematical Functions: §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions Other Notations ‣ §23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions §23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions §23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions §23.20(v) Modular Functions and Number Theory ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions